= Dirichlet Green function for a three-dimensional quarter-space
{c}
{title2=$G(P,Q)=\sum_{i,j=0}^1(-1)^{i+j}/(4\pi|P-Q_{ij}|)$}
For $D=\{x>0,z>0\}\subset\mathbb R^3$, take the <Dirichlet Green function> convention $-\Delta_PG(P,Q)=\delta_Q(P)$. Reflect $Q=(\xi,\eta,\zeta)$ to $Q_{ij}=((-1)^i\xi,\eta,(-1)^j\zeta)$. The displayed image sum vanishes on both faces, has exactly one positive unit source inside $D$, and decays at infinity. Pairwise cancellation proves the boundary conditions, while the three image terms are <harmonic> inside $D$. Its inward source-normal derivative gives the <Poisson kernel of a three-dimensional quarter-space>. The source sign must be kept consistent with <Green's second identity>.
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